How to figure out which words have the same meaning in two different languages?
You are implying that such ideas are novel, and that such tools do not exist. But the idea is very popular, and there are numerous tools.


  We need to write a program that would recognize that a word is connected to other words in the same way in both language. Then it would know those two words must have the same meaning.


You are describing the essence of known natural language processing (NLP) tasks such as word alignment (link words in different languages that have the same meaning) and, of course, machine translation.

While learning a machine translation model, we actually do discover which words (or parts of words, or sequences of words) in different languages have the same meaning.

Here are some concepts I would recommend for further study of this subject:


Word alignment, an example for a well-known and popular tool would be fast_align
Word embeddings, word2vec is a widely used tool
Modern machine translation with sequence-to-sequence models, well-known tools are fairseq, or Sockeye